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Exponentials:
What Is Exponential Change?
Exponential growth and decay both use the idea of changing by a percentage again and again — like increasing or decreasing every year.
General Formula:
Surds:
What Are Surds?
A surd is an irrational root that cannot be simplified into a whole number.
Examples of surds:
√2, √3, √5 (These cannot be simplified into a whole number.)
Examples that are NOT surds:
√4 = 2
√9 = 3
√25 = 5
Practice Problem:
Which of these are surds: √12, √81, √8, √49?
Answer: √12 and √8
Simplifying Surds
We break a surd into two parts: a square number × a leftover.
More Examples:
√50 = √25 × √2 = 5√2
√200 = √100 × √2 = 10√2
√32 = √16 × √2 = 4√2
Adding and Subtracting Surds
You can only add/subtract like surds (same √ number), just like like terms in algebra.
3√2+2√2=5√2 But 3√2+√3=can’t simplify
Practice:
Simplify: 5√3 − 2√3 + √3
Answer: they have the same surd so 3√3 + 1√3 = 4√3
Multiplying Surds
√a×√b=√(a×b)
Example:
√2 × √5 = √10
√3 × √3 = √9 = 3
Rationalizing the Denominator
You’re often asked to “rationalize the denominator” — this means removing any surds from the bottom of a fraction.
CASE 1: When the denominator is a single surd:
CASE 2: When the denominator is in the form a + √b:
You use the conjugate
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